On the theory of the Ranque-Hilsch cooling effect


An extended Bernoulli equation, adapted to turbulent flow, is applied to a model of the rotational flow in the Ranque-Hilsch vortex tube. It is shown that on the basis of this model a reasonable quantitative explanation of the cooling effect can be given. The experimental results of Hilsch and of Elser and Hoch agree for the greater part with this theory.

As to the construction of the apparatus a theoretical interpretation is given of the influence of the size of the diaphragm.

This is a preview of subscription content, access via your institution.


c p :

specific heat at constant pressure

c v :

specific heat at constant volume

M :

Mach number

P :

ηc p /λ = Prandtl number

p :


p 0 :

inlet pressure

p 1 :

pressure at the wall

r :

coordinate in radial direction

r 1 :

tube radius

R′ :

gas constant per unit mass

s :

vector defined by equation (17);s r ,s ϑ ,s z are the components in cylindrical coordinates

T :

absolute temperature (static)

T 0 :

inlet temperature

T 1 :

temperature atr=r 1

T s :

T1+v2/2c p = stagnation temperature

v :

velocity vector;u,v,w are the components in cylindrical coordinates;u 1 andv 1 are the values ofu andv atr =r 1

z :

coordinate in axial direction


η e c p /λ e


dimensionless quantity defined by equation (22)


c p /c v


coefficient of viscosity

η e :

coefficient of eddy viscosity


angular coordinate


thermal conductivity

λ e :

eddy thermal conductivity


mass fraction of cooled gas



ϱ 1 :

value of ϱ atr=r 1






nabla operator


  1. 1)

    Ranque, M. G., J. Phys. Radium4 (1933) 2.

    Google Scholar 

  2. 2)

    Hilsch, R., Z. Naturforsch.1 (1946) 208.

    ADS  Google Scholar 

  3. 3)

    Elser, K., and M. Hoch, Z. Naturforsch.6a (1951) 25.

    ADS  Google Scholar 

  4. 4)

    Haar, D. ter, and H. Wergeland, Forh. Kong. Norske Vid. Selskab.20 (1947) 55.

    Google Scholar 

  5. 5)

    Burkhardt, G., Z. Naturforsch.3a (1948) 40.

    ADS  Google Scholar 

  6. 6)

    Prins, J. A., Ned. T. Natuurk.14 (1948) 241.

    Google Scholar 

  7. 7)

    Ringleb, F., Z. angew. Math. Mech.20 (1940) 185.

    Article  MathSciNet  Google Scholar 

  8. 8)

    Madelung, E., Ann. Phys.43 (1943) 417.

    Article  Google Scholar 

  9. 9)

    Rubusin, M. W., and H. A. Johnson, Trans. Amer. Soc. mech. Eng.71 (1949) 383.

    Google Scholar 

  10. 10)

    Reichardt, H., Z. angew. Math. Mech.20 (1940) 297.

    Article  MathSciNet  Google Scholar 

  11. 11)

    Shepherd, C. B., and C. E. Lapple, Ind. engng. Chem.: 51 (1939) 972;32 (1940) 1246.

    Article  Google Scholar 

  12. 12)

    Eckert, E., and W. Weise, Forsch. Ing. Wes.13 (1942) 246.

    Article  Google Scholar 

  13. 13)

    Ackeret, J., Ric. sci.20 (1950) 1926.

    Google Scholar 

  14. 14)

    Webster, D. S., J. A. S. Refr. Eng.58 (1950) 163.

    Google Scholar 

  15. 15)

    Fulton, C. D., Ibid. 473.

    Google Scholar 

  16. 16)

    16) Roy Mac Gee Jr., Ibid. 974.

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to J. J. Van Deemter.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Van Deemter, J.J. On the theory of the Ranque-Hilsch cooling effect. Appl. sci. Res. 3, 174–196 (1952). https://doi.org/10.1007/BF03184927

Download citation


  • Velocity Profile
  • Mach Number
  • Cooling Effect
  • Vortex Tube
  • Stagnation Temperature